For \(\rho \in (G)_s\)  for some \(s\) ,
we write \(\Xi _s(\rho )=\rho ^{(0)}\otimes \rho _{\Lambda ^{(-)}}\otimes \rho _{\Lambda ^{(+)}}\) .
From [1]} 4.23 (see also [2]} remark 13.24),
it is known that
\(\rho (1)=\frac{|G|_{p^{\prime }}}{|G^{(0)}|_{p^{\prime }}|G^{(-)}|_{p^{\prime }}|G^{(+)}|_{p^{\prime }}}\rho ^{(0)}(1)\rho _{\Lambda ^{(-)}}(1)\rho _{\Lambda ^{(+)}}(1).\) 

Suppose that \((,^{\prime })=(^\epsilon _k,_n)\)  or \((_k,^\epsilon _n)\) 
where both \(k,n\)  are even.
Now we have a commutative diagram
\({\begin{matrix}\rho &\xrightarrow{}& \rho ^{\prime } \\{\scriptstyle  \Xi _s }\downarrow \mathbox{mphantom}{\scriptstyle  \Xi _s }&& \mathbox{mphantom}{\scriptstyle  \Xi _{s^{\prime }} }\downarrow {\scriptstyle  \Xi _{s^{\prime }} }&& \\\rho ^{(0)}\otimes \rho _{\Lambda ^{(-)}}\otimes \rho _{\Lambda ^{(+)}} &\xrightarrow{}& \rho ^{\prime (0)}\otimes \rho _{\Lambda ^{\prime (-)}}\otimes \rho _{\Lambda ^{\prime (+)}} \\\end{matrix}}\) 
We know that \(^{(0)}\simeq ^{\prime (0)}\) , \(\rho ^{(0)}=\rho ^{\prime (0)}\) 
and \(^{(-)}\simeq ^{\prime (-)}\) , \(\rho _{\Lambda ^{(-)}}=\rho _{\Lambda ^{\prime (-)}}\) .
Therefore by (REF ),
we have
\(\deg _q(\underline{\theta }(\rho ))-\deg _q(\rho )=\deg _q(\underline{\theta }(\rho _{\Lambda ^{(+)}}))-\deg _q(\rho _{\Lambda ^{(+)}}) \\+\deg _q(|G^{\prime }|_{p^{\prime }})-\deg _q(|G|_{p^{\prime }})-\deg _q(|G^{\prime (+)}|_{p^{\prime }})+\deg _q(|G^{(+)}|_{p^{\prime }}).\) 

Suppose that \((,^{\prime })=(^\epsilon _k,_n)\) .
Then \(^{(+)}=^\epsilon _{k-2l}\)  and \(^{\prime (+)}=_{n-2l}\)  for some non-negative integer \(l\) .
Hence by Proposition REF , we have
\(\deg _q(\underline{\theta }(\rho _{\Lambda ^{(+)}}))-\deg _q(\rho _{\Lambda ^{(+)}})&= \tfrac{1}{2}(k-2l)(n-2l-(k-2l)+1), \\\deg _q(|G^{\prime }|_{p^{\prime }})-\deg _q(|G|_{p^{\prime }})&= \tfrac{1}{4}n(n+2)-\tfrac{1}{4}k^2, \\-\deg _q(|G^{\prime (+)}|_{p^{\prime }})+\deg _q(|G^{(+)}|_{p^{\prime }})&=-\tfrac{1}{4}(n-2l)(n-2l+21)+\tfrac{1}{4}(k-2l)^2.\) 

Therefore,
\(& \deg _q(\underline{\theta }(\rho ))-\deg _q(\rho ) \\&= \tfrac{1}{2}(k-2l)(n-k+1)+\tfrac{1}{4}n(n+2)-\tfrac{1}{4}k^2-\tfrac{1}{4}(n-2l)(n-2l+2)+\tfrac{1}{4}(k-2l)^2 \\&=\tfrac{1}{2}k(n-k+1).\) 

Suppose that \((,^{\prime })=(_k,^\epsilon _n)\) .
Then \(^{(+)}=_{k-2l}\)  and \(^{\prime (+)}=^\epsilon _{n-2l}\)  for some non-negative integer \(l\) .
Hence by Proposition REF , we have
\(\deg _q(\underline{\theta }(\rho _{\Lambda ^{(+)}}))-\deg _q(\rho _{\Lambda ^{(+)}})&= \tfrac{1}{2}(k-2l)(n-2l-(k-2l)-1), \\\deg _q(|G^{\prime }|_{p^{\prime }})-\deg _q(|G|_{p^{\prime }})&= \tfrac{1}{4}n^2-\tfrac{1}{4}k(k+2), \\-\deg _q(|G^{\prime (+)}|_{p^{\prime }})+\deg _q(|G^{(+)}|_{p^{\prime }})&=-\tfrac{1}{4}(n-2l)^2+\tfrac{1}{4}(k-2l)(k-2l+2).\) 

Therefore
\(& \deg _q(\underline{\theta }(\rho ))-\deg _q(\rho ) \\&= \tfrac{1}{2}(k-2l)(n-k-1)+\tfrac{1}{4}n^2-\tfrac{1}{4}k(k+2)-\tfrac{1}{4}(n-2l)^2+\tfrac{1}{4}(k-2l)(k-2l+21) \\&= \tfrac{1}{2}k(n-k-1).\) 


Suppose that \((,^{\prime })=(_k,_n)\)  where \(k\)  is odd and \(n\)  even;
or \((_k,_n)\)  where \(k\)  is even and \(n\)  is odd.

Suppose that \((,^{\prime })=(_k,_n)\)  where \(k\)  is odd and \(n\)  is even.
Now we have a commutative diagram
\({\begin{matrix}\rho &\xrightarrow{}& \rho ^{\prime } \\{\scriptstyle  \iota \circ \Xi _s }\downarrow \mathbox{mphantom}{\scriptstyle  \iota \circ \Xi _s }&& \mathbox{mphantom}{\scriptstyle  \Xi _{s^{\prime }} }\downarrow {\scriptstyle  \Xi _{s^{\prime }} }&& \\\rho ^{(0)}\otimes \rho _{\Lambda ^{(+)}}\otimes \rho _{\Lambda ^{(-)}} &\xrightarrow{}& \rho ^{\prime (0)}\otimes \rho _{\Lambda ^{\prime (-)}}\otimes \rho _{\Lambda ^{\prime (+)}} \\\end{matrix}}\) 
We know that \(^{(0)}\simeq ^{\prime (0)}\) , \(\rho ^{(0)}=\rho ^{\prime (0)}\) 
and \(^{(-)}\simeq ^{\prime (+)}\) , \(\rho _{\Lambda ^{(-)}}=\rho _{\Lambda ^{\prime (+)}}\) .
Therefore by (REF ),
we have
\(\deg _q(\underline{\theta }(\rho ))-\deg _q(\rho )=\deg _q(\underline{\theta }(\rho _{\Lambda ^{(+)}}))-\deg _q(\rho _{\Lambda ^{(+)}}) \\+\deg _q(|G^{\prime }|_{p^{\prime }})-\deg _q(|G|_{p^{\prime }})-\deg _q(|G^{\prime (-)}|_{p^{\prime }})+\deg _q(|G^{(+)}|_{p^{\prime }}).\) 
Then \(^{(+)}=_{k-2l-1}\)  and \(^{\prime (-)}=^\epsilon _{n-2l}\)  for some non-negative integer \(l\) .
Hence by Proposition REF , we have
\(\deg _q(\underline{\theta }(\rho _{\Lambda ^{(-)}}))-\deg _q(\rho _{\Lambda ^{(-)}})&= \tfrac{1}{2}(k-2l-1)(n-2l-(k-2l-1)-1), \\\deg _q(|G^{\prime }|_{p^{\prime }})-\deg _q(|G|_{p^{\prime }})&= \tfrac{1}{4}n(n+2)-\tfrac{1}{4}(k-1)(k+1), \\-\deg _q(|G^{\prime (-)}|_{p^{\prime }})+\deg _q(|G^{(+)}|_{p^{\prime }})&=-\tfrac{1}{4}(n-2l)^2+\tfrac{1}{4}(k-1-2l)(k+1-2l).\) 

Therefore
\(\deg _q(\underline{\theta }(\rho ))-\deg _q(\rho )&= \tfrac{1}{2}(k-2l-1)(n-k)+\tfrac{1}{4}n(n+2)-\tfrac{1}{4}(k-1)(k+1) \\&\qquad \qquad \qquad \qquad -\tfrac{1}{4}(n-2l)^2+\tfrac{1}{4}(k-1-2l)(k+1-2l) \\&= \tfrac{1}{2}k(n-k+1).\) 

Suppose that \((,^{\prime })=(_k,_n)\)  where \(k\)  is even and \(n\)  is odd.
Now we have a commutative diagram
\({\begin{matrix}\rho &\xrightarrow{}& \rho ^{\prime } \\{\scriptstyle  \Xi _s }\downarrow \mathbox{mphantom}{\scriptstyle  \Xi _s }&& \mathbox{mphantom}{\scriptstyle  \iota \circ \Xi _{s^{\prime }} }\downarrow {\scriptstyle  \iota \circ \Xi _{s^{\prime }} }&& \\\rho ^{(0)}\otimes \rho _{\Lambda ^{(-)}}\otimes \rho _{\Lambda ^{(+)}} &\xrightarrow{}& \rho ^{\prime (0)}\otimes \rho _{\Lambda ^{\prime (+)}}\otimes \rho _{\Lambda ^{\prime (-)}} \\\end{matrix}}\) 
We know that \(^{(0)}\simeq ^{\prime (0)}\) , \(\rho ^{(0)}=\rho ^{\prime (0)}\) 
and \(^{(+)}\simeq ^{\prime (-)}\) , \(\rho _{\Lambda ^{(+)}}=\rho _{\Lambda ^{\prime (-)}}\) .
Therefore by (REF ),
we have
\(\deg _q(\underline{\theta }(\rho ))-\deg _q(\rho )=\deg _q(\underline{\theta }(\rho _{\Lambda ^{(-)}}))-\deg _q(\rho _{\Lambda ^{(-)}}) \\+\deg _q(|G^{\prime }|_{p^{\prime }})-\deg _q(|G|_{p^{\prime }})-\deg _q(|G^{\prime (+)}|_{p^{\prime }})+\deg _q(|G^{(-)}|_{p^{\prime }}).\) 
Then \(^{(-)}=^\epsilon _{k-2l}\)  and \(^{\prime (+)}=_{n-2l-1}\)  for some non-negative integer \(l\) .
Hence by Proposition REF , we have
\(\deg _q(\underline{\theta }(\rho _{\Lambda ^{(-)}}))-\deg _q(\rho _{\Lambda ^{(-)}})&= \tfrac{1}{2}(k-2l)(n-2l-1-(k-2l)+1), \\\deg _q(|G^{\prime }|_{p^{\prime }})-\deg _q(|G|_{p^{\prime }})&= \tfrac{1}{4}(n-1)(n+1)-\tfrac{1}{4}k(k+2), \\-\deg _q(|G^{\prime (+)}|_{p^{\prime }})+\deg _q(|G^{(-)}|_{p^{\prime }})&=-\tfrac{1}{4}(n-1-2l)(n+1-2l)+\tfrac{1}{4}(k-2l)^2.\) 

Therefore
\(\deg _q(\underline{\theta }(\rho ))-\deg _q(\rho )&= \tfrac{1}{2}(k-2l)(n-k)+\tfrac{1}{4}(n-1)(n+1)-\tfrac{1}{4}k(k+2) \\&\qquad \qquad \qquad \quad -\tfrac{1}{4}(n-1-2l)(n+1-2l)+\tfrac{1}{4}(k-2l)^2 \\&=\tfrac{1}{2}k(n-k-1).\) 


